Optimal. Leaf size=60 \[ -\frac{a^2 \sqrt [4]{a-b x^4}}{b^3}-\frac{\left (a-b x^4\right )^{9/4}}{9 b^3}+\frac{2 a \left (a-b x^4\right )^{5/4}}{5 b^3} \]
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Rubi [A] time = 0.0335839, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {266, 43} \[ -\frac{a^2 \sqrt [4]{a-b x^4}}{b^3}-\frac{\left (a-b x^4\right )^{9/4}}{9 b^3}+\frac{2 a \left (a-b x^4\right )^{5/4}}{5 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{\left (a-b x^4\right )^{3/4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{(a-b x)^{3/4}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a-b x)^{3/4}}-\frac{2 a \sqrt [4]{a-b x}}{b^2}+\frac{(a-b x)^{5/4}}{b^2}\right ) \, dx,x,x^4\right )\\ &=-\frac{a^2 \sqrt [4]{a-b x^4}}{b^3}+\frac{2 a \left (a-b x^4\right )^{5/4}}{5 b^3}-\frac{\left (a-b x^4\right )^{9/4}}{9 b^3}\\ \end{align*}
Mathematica [A] time = 0.0184807, size = 40, normalized size = 0.67 \[ -\frac{\sqrt [4]{a-b x^4} \left (32 a^2+8 a b x^4+5 b^2 x^8\right )}{45 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 37, normalized size = 0.6 \begin{align*} -{\frac{5\,{b}^{2}{x}^{8}+8\,ab{x}^{4}+32\,{a}^{2}}{45\,{b}^{3}}\sqrt [4]{-b{x}^{4}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02536, size = 68, normalized size = 1.13 \begin{align*} -\frac{{\left (-b x^{4} + a\right )}^{\frac{9}{4}}}{9 \, b^{3}} + \frac{2 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} a}{5 \, b^{3}} - \frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{2}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74533, size = 85, normalized size = 1.42 \begin{align*} -\frac{{\left (5 \, b^{2} x^{8} + 8 \, a b x^{4} + 32 \, a^{2}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{45 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.44526, size = 70, normalized size = 1.17 \begin{align*} \begin{cases} - \frac{32 a^{2} \sqrt [4]{a - b x^{4}}}{45 b^{3}} - \frac{8 a x^{4} \sqrt [4]{a - b x^{4}}}{45 b^{2}} - \frac{x^{8} \sqrt [4]{a - b x^{4}}}{9 b} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 a^{\frac{3}{4}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12793, size = 77, normalized size = 1.28 \begin{align*} -\frac{5 \,{\left (b x^{4} - a\right )}^{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} - 18 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} a + 45 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{2}}{45 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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